Unrestricted Polar Quantization Research Articles

Design of General Entropy-Constrained Successively Refinable Unrestricted Polar Quantizer

This paper presents an algorithm for the optimal design of general entropy-constrained successively refinable unrestricted polar quantizer, i.e., with arbitrary number $L$ of refinement levels, for bivariate circularly symmetric sources. The optimization problem is formulated as the minimization of a weighted sum of distortions and entropies for the scenario where the magnitude quantizers’ thresholds are confined to a predefined finite set. The proposed solution algorithm is globally optimal. It involves $L$ stages, where each stage corresponds to an unrestricted polar quantizer (UPQ) level, and includes solving the minimum-weight path problem for multiple node pairs in a series of weighted directed acyclic graphs. Additionally, we derive an upper bound $P^{(l)}_{\max }, l\in [1:L]$ , on the possible number of phase levels in any phase quantizer of the $l$ -th level UPQ, which grows linearly with $l$ . The time complexity of the proposed approach is $O(L^{2}K^{3} P^{(1)}_{\max })$ , where $K$ is the cardinality of the predefined set of possible magnitude thresholds. Finally, the experimental results for $L=3$ demonstrate the effectiveness in practice of the proposed scheme.

Design of Successively Refinable Unrestricted Polar Quantizer

This paper addresses the design of two-stage successively refinable unrestricted polar quantizers for bivariate circularly symmetric sources in the entropy-constrained and fixed-rate cases. The proposed solutions are globally optimal when the thresholds of the magnitude quantizers are confined to finite discretizations of the interval $\left[0,\infty\right)$ . The algorithm developed for the entropy-constrained case involves a series of stages, including solving the minimum-weight path problem for multiple node pairs in certain weighted directed acyclic graphs. The asymptotical time complexity is ${O}({K}_{1}{K}_{2}^{2}{P}_{max})$ , where ${K}_{1}$ and ${K}_{2}$ are the sizes of the sets of possible magnitude thresholds of the coarse and refined unrestricted polar quantizers (UPQs), respectively, while ${P}_{max}$ is an upper bound on the number of phase levels in any phase quantizer of the coarse UPQ. The solution algorithm for the fixed-rate case is based on solving a succession of dynamic programming problems for multiple coarse quantizer bins. The time complexity in the fixed-rate case amounts to ${O}({K}_{1}{K}_{2}{N}^{2}{N}_{1})$ , where ${N}_{1}$ is the number of cells of the coarse UPQ and $N$ is the ratio between the number of bins of the fine and coarse UPQs. The extensive experimental results on a bivariate circularly symmetric Gaussian source show the effectiveness of the proposed schemes.

An efficient iterative algorithm for designing an asymptotically optimal modified unrestricted uniform polar quantization of bivariate Gaussian random variables

In this paper, a more efficient and a more accurate algorithm is developed for designing asymptotically optimal unrestricted uniform polar quantization (UUPQ) of bivariate Gaussian random variables compared to the existing algorithms on this subject. The proposed algorithm is an iterative one defining the analytical model of asymptotically optimal UUPQ in only a few iterations. The UUPQ model is also improved via optimization of the last magnitude reconstruction level so that the mean squared error (MSE) is minimal. Moreover, for the straightforward performance assessment of our analytical UUPQ model an asymptotic formula for signal to quantization noise ratio (SQNR) is derived, which is reasonably accurate for any rate (R) greater than or equal to 2.5 bits/sample. It is demonstrated empirically that our asymptotically optimal UUPQ model outperforms the previous UUPQ models in terms of SQNR. Eventually, the transition from the analytical to the practically designed UUPQ model, as an important aspect in quantizer design, is considered in the paper and, as a result, a novel method to achieve this is provided. The proposed method is applicable to the practical design of any unrestricted polar quantization.

Design of Optimal Fixed-Rate Unrestricted Polar Quantizer for Bivariate Circularly Symmetric Sources

This letter presents an algorithm for the design of fixed-rate unrestricted polar quantizer (FUPQ) for bivariate circularly symmetric sources. The proposed algorithm is globally optimal for the class of FUPQs with the magnitude quantizer thresholds restricted to some predefined finite set. The solution algorithm is based on dynamic programming, which is further accelerated by exploiting the monotonicity property of the cost function. The time complexity of the accelerated algorithm is $O(KN^2)$ , where $N$ is the number of target qunatizer levels and $K$ is the size of the predefined set of possible thresholds. The experimental results show that our approach outperforms the previous tractable designs when the total number of quantizer levels ranges between 25 and 256.

Design of Optimal Entropy-Constrained Unrestricted Polar Quantizer for Bivariate Circularly Symmetric Sources

This paper proposes an algorithm for the design of entropy-constrained unrestricted polar quantizer (ECUPQ) for bivariate circularly symmetric sources. The algorithm is globally optimal for the class of ECUPQs with magnitude quantizer thresholds confined to a finite set. The optimization problem is formulated as the minimization of a weighted sum of distortion and entropy, and the proposed solution is based on modeling the problem as a minimum-weight path problem in a certain weighted directed acyclic graph. Each graph edge corresponds to a possible magnitude quantizer bin and computing its weight involves solving another optimization problem. We develop a fast strategy for evaluating all edge weights, leading to a $O(K^{2}+\textit {KP}_{\mathrm{ max}})$ time solution algorithm, where $K$ is the size of the set of possible magnitude thresholds and $P_{\mathrm{ max}}$ is the maximum number of phase levels. The practical performance of the proposed algorithm is assessed for a bivariate circularly symmetric Gaussian source, at rates ranging from 0.5 to 6 bits/sample. Our results demonstrate that the proposed approach achieves performance very close to the asymptotically optimal ECUPQ at all rates, while at low rates it significantly outperforms all previous UPQ schemes. Notably, peak improvement of 0.755 dB can be achieved for rates below 2.5.

Multiproduct Uniform Polar Quantizer

The aim of this paper is to reduce the complexity of the unrestricted uniform polar quantizer (UUPQ), keeping its high performances. To achieve this, in this paper we propose the multiproduct uniform polar quan- tizer (MUPQ), where several consecutive magnitude levels are joined in segments and within each segment the uni- form product quantization is performed (i.e. all levels within one segments have the same number of phase levels). MUPQ is much simpler for realization than UUPQ, but it achieves similar performances as UUPQ. Since MUPQ has low complexity and achieves much better performances than the scalar uniform quantizer, it can be widely used instead of scalar uniform quantizers to im- prove performances, for any signal with the Gaussian distribution.

OPTIMAL POLAR QUANTIZATION OF COMPLEX VARIABLES WITH CIRCULARLY SYMMETRIC DENSITIES

In this paper we consider quantization of complex variables and mean-square error (MSE). The best polar quantizer is Wilson’s unrestricted polar quantizer (UPQ) [1]. The MSE minimization is constrained only by the total number of quantization points, N. Our method is different from Wilson’s algorithm [1] that has predetermined number of points Mi at each magnitude level i, 1?i?L, which makes it impractical for large number of points. In our approach, we consider MSE as a function of the vector M= L i i M ? ? 1 ) (whose elements are numbers of phase quantization levels at each magnitude level. The Wilson's method finds the optimal quantizer in such a way that the decision and reconstruction levels r, m are iterative found for each combination M, while the optimal combination is found by searching all combinations. Wilson's algorithm cannot be applied for middle and great N. The asymptotic analysis of the polar quantizers with circular symmetric densities is given in [2]. This analysis is approximate and cannot be applied for any number of points and for great N, which will be shown in this paper. We define the extension of the MSE over RL (denoted by MSE(P)). We prove the convexity of this function and show an efficient way to find M= L i i M ? ? 1 ) (by Popt. Our algorithm consists of two main iterative processes. The first iterative process finds Popt, ropt, mopt with ? accuracy, while the second iterative process determines Mopt, mopt, ropt using Popt as the starting value. This paper eliminates incompleteness from [1] and [2]. We also give an example of the quantizer construction for a Gaussian source. The authors see their work as a contribution in knowing the best possible solution in these classes of problems and also a possibility of applying the technique exposed inhere on other classes of problems and on larger dimensions.

Variance Mismatch Analysis of Unrestricted Polar Quantization for Gaussian Source

In this letter, asymptotic formulas are derived for signal to quantization noise ratio (SQNR) of two unrestricted polar quantizers (UPQs) for the case where designed-for and applied-to Gaussian sources have different variances. It is shown that the limitation in the variance mismatch analysis observed in one of the UPQs can be overcome by the constrained application of asymptotic approximations and by optimizing the last magnitude reconstruction level subsequently. The derived formulas are a useful analytical tool for determining the effect of variance mismatch on SQNR in unrestricted polar quantization for Gaussian source.

Design of Asymptotically Optimal Unrestricted Polar Quantizer for Gaussian Source

In this letter, in order to outperform the existing method for unrestricted polar quantizer (UPQ) design in terms of signal to quantization noise ratio, the asymptotic approximations of Rayleigh distributed function are applied to all magnitude regions of the UPQ, except to the last one. Given the constraint, the UPQ is designed to provide the minimum of the asymptotic mean-squared error distortion for the Gaussian source of unit variance. The effects of this constraint are studied for different bit rates R. The accuracy of the derived formulas is assessed and the reasonable accuracy is observed for R ≥ 2.5 bit/sample.

Variable-length coding for performance improvement of asymptotically optimal unrestricted polar quantization of bivariate Gaussian source

This paper considers the application of variable-length coding using two unrestricted polar quantizers (UPQs) for performance improvement of unrestricted polar quantization for bivariate Gaussian source. We propose the use of two UPQs, both designed for the bivariate Gaussian source of unit variance, having different sizes of codebooks and different optimal compressor functions. We show that the fixed-rate UPQ is a subset of our model, and we perform rigorous optimization procedure in order to obtain optimal parameter values maximizing the signal to quantization noise ratio (SQNR) for the given average bit rate. In addition, we study how gain in SQNR over the fixed-rate UPQ depends on the average bit rate and we show that the gain ranges from 0.619dB to 0.869dB depending on the average bit rate. Discussion is also provided about the proposed quantizer complexity and its performance in comparison to Shannon limit. The proposed UPQ provides a sophisticated choice of average bit rate compared to the fixed-rate UPQ. Features of the proposed quantizer indicate that the obtained model should be of high theoretical and practical significance.

The general design of asymptotic unrestricted polar quantizers with square cells

The aim of this paper is the presentation of the designing of asymptotic unrestricted polar quantizers with square cells, since it is known that square cells give minimal moment of inertia, which leads to minimization of distortion. Until now, polar quantizers with square cells have been designed only for the optimal companding function and their performances have been analyzed only for the stationary variance. In this paper, the design is done in a general manner, i.e. it is valid for any companding function and performances are analyzed in the wide range of variances. After that, this general design is applied for the logarithmic μ -law companding function. It is important that expressions for the numbers of magnitude and phase levels are obtained in the closed form, which simplifies the design. It is shown that the proposed polar quantizer has better performances (more than 3 dB higher the maximal and the average SQNR (signal-to-quantization noise ratio)) than the corresponding scalar quantizer with μ -law. Simulation is performed, and it is shown that theoretical and simulation results are matched very well. It is shown that the proposed polar quantizer can be used both for stationary and non-stationary signals, choosing the appropriate value of μ . This quantizer can be widely used, for all signals with Gaussian distribution. • The design of asymptotic unrestricted polar quantizer with square cells was done. • The design was done in general manner, i.e. for any compression function. • The design was done for variable variance. • Using μ -law companding function, robust polar quantizers were designed. • Very good performances are achieved and proved by simulation.

Asymptotic Analysis of Switched Uniform Polar Quantization for Memoryless Gaussian Source

This letter performs an asymptotic analysis of the switched uniform polar quantizer (SUPQ) composed of k asymptotically optimal unrestricted uniform polar quantizers designed for the memoryless Gaussian source. The closed-form formulas are derived for signal to quantization noise ratio (SQNR) and the number of phase levels of the quantizers constituting the SUPQ. It is studied how SQNR depends on the variance mismatch and the number of quantizers k. It is shown that with a log-uniform distribution of the variances for which the quantizers constituting the SUPQ are designed one can reduce the variance range of average-taking of SQNR.

An iterative method for optimal resolution‐constrained polar quantizer design

PurposeThe purpose of this paper is to address the problem of polar quantization optimization. Particularly, the aim is to find the method for the optimal resolution‐constrained polar quantizer design.Design/methodology/approachThe new iterative algorithm for determination of the optimal decision and representation magnitude levels and algorithm for optimization of number of phase cells within each magnitude level, is proposed.FindingsAt high rates, the new optimal polar quantizer outperforms the optimal polar compander for 0.2 dB, while the more significant gain should be expected at lower rates. In this paper, in order to enable practical implementation of quantizer model, algorithm which transforms real values for the optimal numbers of phase cells within magnitude levels into integer ones is also proposed. Moreover, the approximate closed form of signal‐to‐quantization ratio is derived.Practical implicationsSince circularly symmetric sources and complex presentation of signals arise in numerous applications, it can be concluded that the usage area of the suggested proposal is very wide (audio coding, image coding, spectral phase coding, synthetic aperture radars systems, coding of the discrete Fourier transform).Originality/valueIt should be emphasized that in contrast to earlier work, where models have been designed under high‐rate assumption, the obtained nonuniform unrestricted polar quantizer is optimal for all rates.

Embedded Polar Quantization

Embedded polar quantization can be useful for progressive transmission of circularly symmetric data, e.g., for fine-grain scalable coding of parametric audio. Sets of constrained-resolution embedded quantizers are built recursively by successive refinement processes, that are detailed for strict polar quantization and unrestricted polar quantization. The quadratic error minimization problem is solved using equations similar to those of Max, and the refinement algorithm can, in the unrestricted case, be simplified using a high-rate approximation. For Gaussian data, comparisons with reference non-embedded quantizers show that the embedding property comes at an often negligible cost in terms of rate-distortion performance.

Multivariate Block Polar Quantization

We introduce multivariate block polar quantization (MBPQ). MBPQ minimizes the weighted squared-error distortion for a set of complex variables representing one block of a signal under a resolution constraint for the entire block. MBPQ performs below the lower bound for classical bivariate quantization, both for Gaussian complex variables and for complex variables found from sinusoidal analysis of audio data. Still, it is of similar complexity as traditional polar quantizers. In the case of audio data, we found a performance gain of about 2.5 dB over the best performing conventional resolution-constrained polar quantization (an extension of unrestricted polar quantization).

Entropy-constrained polar quantization and its application to audio coding

In this work, we present a new method for quantization of sinusoidal amplitudes and phases, and apply the method to sinusoidal coding of speech and audio signals. The method is based on unrestricted polar quantization, where phase quantization accuracy depends on amplitude. Amplitude and phase quantizers are derived under an entropy (average rate) constraint using high-rate assumptions. First, we derive optimal quantizers for one sinusoid and a mean-squared error distortion measure. We provide a detailed analysis of entropy-constrained unrestricted polar quantization, showing its high performance and practicality even at low rates. Second, we find optimal quantizers for a set of sinusoids that model a short segment of an audio signal. The optimization is performed using a weighted error measure that can account for the masking effect in the human auditory system. We find the optimal rate distribution between sinusoids, as well as the corresponding optimal amplitude and phase quantizers, based on the perceptual importance of sinusoids defined by masking. The new method is used in an audio-coding application and is shown to significantly outperform a conventional sinusoidal quantization method where phase quantization accuracy is identical for all sinusoids.

A vector quantizer for the Laplace source

The low complexity, nearly optimal vector quantizer (VQ) is a generalization of T. R. Fischer's (1986) pyramid VQ and is similar in structure to the unrestricted polar quantizers previously presented for the independent Gaussian source. An analysis of performance is presented with results for both the product code pyramid VQ and the unrestricted version. This analysis, although asymptotic in nature, helps to demonstrate the performance advantages of the VQ. Implementation issues of the VQ are discussed. Nonasymptotic results are considered. In particular, the author presents an approximate design algorithm for finite bit rate and demonstrates the usefulness of this VQ through several example designs with Monte Carlo simulations of performance. For the restricted form (the pyramid VQ), the author provides further implementational information and low dimension analytical results. >

Asymptotic performance of unrestricted polar quantizers (Corresp.)

Unrestricted polar quantizers (UPQ's) were previously introduced to overcome the weak mean-squared error performance of polar coordinate quantizers for the independent bivariate Gaussian source when the number of levels was small ( bits /dimension). An asymptotic ntean-squared error analysis of the UPQ's with a general circularly symmetric source is presented. The performance of the UPQ's is shown to be asymptotically within 0.17 dB of that of the optimum bivariate quantizer. The results are extended to dimensions greater than two. Examples for the independent Gaussian source are included.